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6. The third and last Letter of the second Word, also the third Letter of the third Word, are the same with the second Letter of the first Word; hence the Letters will stand thus,

Soli Deo Glo * i *.

7. The Sum of the Indices of the fourth Letter of the third Word, and the fixth or last Letter of the fame Word, being added to their Product is 35; and the Difference of their Squares is 288; the Index of the last Letter being the least.

Put a = the greater, and e= the lesser Index, as before.

Then 11ac + a +=3} by the Data.

And 2a a

ie= 288


3ae te= 35 30+144 = 35, for exa+1=aito


1225 - 700 taa 49_1 sle e =

a a +24+1 2 + 5 Gaa=288+

1225 - 700 taa

aa + 2a +1 6 xa a &c. 7

+2 a taa = 288 a +576 + 288

1+1225 — 700 taa
7+ Bla+ +2a — 288 a a- 506 a=1513

This last Equation being resolved according to the Method which shall be thewed in the next Chapter, it will be a = 17 it's Letter ; and from the 4th Stepe=

35 – a

=1, the Index of

a +I the Letter a. Then these two Letters being placed according to the Data above, are all that are required by the Enigma to compleat these Words.

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The Solution of adfected Equations in Numbers." BEFORE we proceed to the Solution

of Adfected Equations, it may not be amiss ta shew the Investigation (or Invention) of those Theorems' or Rules for extracting the Roots of Simple Powers, made use of in Chapter 11. Part 1. I shall here make Choice of the fame Letters to represent the Numbers both given and fought in my Compendium of Algebra.,

G, always denote the given Resolvend. Viz. Let

{ as

may be, whether it be greater or less. {

the unknown Part of the Root fought by

which r is to be either increased or decreased. Then ifr be any Number less than the true Root, it will be rtes the Root sought. But if , be taken greater than the true Root, it will then berce= the Root fought. And put D for the Dividend that is produced from G, after it is lefened and divided by r, &c. (into the Co-ëfficients of Adfected Equations) according as the Nature of the Root requires. These Things being premised, we may proceed to raising the Theorems.

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1. FOR the Square Root, viz. a a=G. Quære a.

Let irtea
I us 279 to zrotec='aa =G
2rr32retu=Gtr. Call it D, viz. D=Gr.


This thews ift the Method of
Then 4

El extracting the Square Root,
2r +

Sect. 5. Chap. 11. Part 1.

Gr 3=2 5.refine=


D Which gives this Theorem

tid The Arithmetical Operations of both these Theortms, you have in the Examples of Section 2. Page 126, to which I refer





the Learner, fuppofing him by this Time to understand them without any more words than what is there expreft.

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II. To extract the Cube Root ; viz. ana=G. Quære a.

Letfifr tie=a, fuppofing - less than the true Root,

I | 2 orrt3rret3r11 tere=aaa =G 2rrrl 33rret376eteee=Gger

Gro 3+371 4retoot

39 31 eee Let be rejected or cast off, as being of small Value; then it

will be retu=D, which gives this following


By this Theorem or Rule, the ist and 2d Examples in Cafe 1.
Page 132, are performed; the which being compared with this
Theorem may be easily understood.

Again, Suppose aaa=G, as before, and let r be taken greater than the true Root. Then

Seee being reject. IQ 2 rrr-3rret3ru==Gled as before. 2 + 33100-3riirror-G

rpg-G 3+371 47

-D 37

D Which gives this Theorem

By this Theorem the third Example in Case 2. Page 133, is performed.


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III. To extract the Biquadrate Root ; viz. 6* =G, Quære a.
Let! 11-e a fuppofing - lels than juft.

2!** +4rrre torre=a*=G{ rejecting all the Pow-
3'477ret'irree=Gqtlers of onbove ce.



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By this Theorem the Biquadrate Root of any Number may be extracted. But, as I have already faid, Page 134, those Extractions may be very well performed by two Extractions of the Square Root. Vide Example, Page 135.

IV. To extract the Surfolid Root, viz. a'=G. Quære a.

If y be taken less than just, then r te=a, as before, and G-75 = D, which gives this Theorem

By this Theorem the Sursolid Root, Example 1. Page 136, is extracted. But if r be taken greater than just; then ri=a, and g3

D = D, which gives this Theorem 5 m this last Theorem the Example in Page 137 is performed.

I presume it needless to pursue the raising of those Theorems, for extracting the Roots of Simple Powers, any further; because the Method of doing it is general, how high soever they are; and therefore it may be easily understood by what is already done.


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SECT. 2. Notwithstanding I have already shewed the Solution of Qua

dratick Equations, two several Ways, viz. by casting off the lowest Term; and by compleating the Square, vide Section 2. Page 195, Sc. Yet it may not be amiss to shew, how those Equations may be resolved into Numbers by this universal Method of continued Series; wherein, if the first r be taken equal to the first true Root, or fingle Side of the Resolvend; and every single Value of e (as it becomes found) be still added to it, for a new r, then those Roots may be extracted without repeating a, fecond Operation, as before in the fingle Powers.

Caje 1. Let a a+ 2 ba=G. It is required to find the Value of a.

Puil 11rtera
IQ? 2 rr +2rite=aa
12b 32 br +2be2ba

2+3) 47r+2br +2re+2bc teezaa+2ba=G 4--1r &c. 52 rit z betee=G-rs-2br 5 • 21 6lr e tbet iee=;


D Which gives this Theorim 7+5+


Suppose -b = 364, and G=38692865: Ifr=6000, then įr = 36000oco, and 2 br = 4368000. But 36000000 + 4368000 = 40368000 7 38692865 =G. Therefore the firit

6000. Let r= 5000, then

19346432,5 =*G
b= 364 -1432000, =irrtbr
iftr +b=5364 5026432,5=D (800 = 6
tie= 400

1 Divisor 5764)

41523 ( 60 = 2dr +b6164

37164 tie= 30

4359 ( 2 Divifor 6194)

3d r +b = 6224

867 = 2

3 Divisor 6227,5
Firft 55909} = 5867 =a as was required.

te 867
Cafe 2. If a 2 2 ba=G, then proceeding as above, there

— will arise this Theorem

=e, &c. And in Cafe 3, rbtie

D viz. 2 ba-aa=G, you will have this Theorem


.e &c. as above.

I think it needless to trouble the Reader with the Work of these. two Theorems in Numbers; because if the last Example of Case 1, be understood, the other will be easy. Not but that the Method of compleating the Square is very ready and easy, as you may observe by the Work in several Questions of this Chapter.

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SECT. 3.

N the Solution of all Adfected Equations, that are above (or

higher than) Quadraticks, it will be the best way to take r= the next nearest Root of the Equation : And then it will be rte =a, if r be less than just; or we=a ifr be greater than juft (as at the Beginning of this Chapter). And all the Powers of the unknown Part of the Root, (viz. e) above it's Square (e e) are to be rejected or cast off, as before in raising the Theorems for the


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